318 XV NORMED ALGEBRAS AND SPECTRAL THEORY
(b) If x, y e A are such that
\\c**y e-t*\\ ^ k\\y\\ (k a constant >0)
for all £ 6 C, show that xy = yx. (Use Liouville's theorem and the Taylor expansion
of the function £ t-te^ye"** at £ = 0.)
(c) If \\x2\\ = ||je||2 for all x e A, show that A is commutative. (Observe that we have
p(x)= \\x\\ for all xeA, and apply this relation to e^ye'1**; then use (a) and (b).)
(d) Show that, if there exists a constant k > 0 such that \\yx\\ <J k\\xy\\ for all x,y
in A, then A is commutative. (Apply the inequality with e~**x in place of x and e;*y
in place of y, and use (b).)
(e) Let a e A be such that
for all £ e C and all x e A. Prove that a belongs to the center of A. (Prove that, for
each f E C and all integers n^n0 (where n0 depends on £)»
for all y e A; then let n tends to •+• oo and use (b).)
15. Let A be a Banach algebra with unit element e, and let C be a commutative Banach
subalgebra containing e. Show that there exists a commutative Banach subalgebra
B of A containing C such that SpB(x) = SPA (x) for each x e C.
3. CHARACTERS AND SPECTRUM OF A COMMUTATIVE BANACH
ALGEBRA. THE GELFAND TRANSFORMATION
Let A be a commutative algebra (over C). A character of A is defined
to be any homomorphisrn x °f the algebra A into C which is not identi-
cally zero. Because #(1*) = %x(x) for all scalars A, this condition is
equivalent to #(A) = C. If A has a unit element e ^ 0, then %(e) ^ 0
(otherwise x(x) =x(ex) = x(e)x(x) — 0 for all x e A); it follows that x(e) = 1>
because %(e)2 =^(e2) = x(e)<
An ideal m ^ A in A is said to be maximal if there exists no ideal n such
that m T»£ n, n ^ A and m c n. If A has a unit element, m is maximal if and
only if A/m is a (commutative) j?eW.
(15.3.1) Let A be a commutative Banach algebra with unit element e =£ 0.
(i) For each x e A and each character x of A, we have x(x) G SpA(:x;).
(ii) Every character x of A is a continuous linear form with norm 1.
(iii) The mapping XI~>X~1(^) ^ a bijection of the set of characters of A
onto the set of maximal ideals of A (which are therefore closed).
Assertion (i) is a particular case of (15.2.8(i)). It follows (15.2.4(iii)) that
\x(x)\ ^ \\x\\, which shows (5.5.1) that ^ is a continuous linear form with
norm gl; and since %(e) = 1 and \\e\\ = 1, we have \\x\\ = 1. Finally, since(c).)